Plane waves can propagate in any direction. Any superposition of these
waves, for all possible
, is also a solution to the wave
equation. However, recall that
and
are not
independent, which restricts the solution in electrodynamics somewhat.

To get a feel for the interdependence of
and
, let's
pick
so that e.g.:

(9.22)

(9.23)

which are plane waves travelling to the right or left along the -axis
for any complex
,
. In one
dimension, at least, if there is no dispersion we can construct a
fourier series of these solutions for various that converges to any
well-behaved function of a single variable.

[Note in passing that:

(9.24)

for arbitrary smooth and is the most general solution of
the 1-dimensional wave equation. Any waveform that preserves its
shape and travels along the -axis at speed is a solution to the
one dimensional wave equation (as can be verified directly, of course).
How boring! These particular harmonic solutions have this form (verify
this).]

If there is dispersion (where the velocity of the waves is a function of
the frequency) then the fourier superposition is no longer stable and
the last equation no longer holds. Each fourier component
is still an exponential, but all the velocities of the fourier
components are different. As a consequence, any initially prepared wave
packet spreads out as it propagates. We'll look at this shortly
(in the homework) in some detail to see how this works for a very simple
(gaussian) wave packet but for now we'll move on.

Note that
and
are connected by having to satisfy
Maxwell's equations even if the wave is travelling in just one direction
(say, in the direction of a unit vector ); we cannot choose
the wave amplitudes separately. Suppose

where
,
, and are constant
vectors (which may be complex, at least for the moment).

Note that applying
to these solutions in the HHE
leads us to:

(9.25)

as the condition for a solution. Then a real leads to the plane wave solution indicated above, with
, which is the most familiar form of the solution
(but not the only one)!

This has mostly been ``mathematics'', following more or less directly
from the wave equation. The same reasoning might have been applied to
sound waves, water waves, waves on a string, or ``waves'' of
nothing in particular. Now let's use some physics and see what it
tells us about the particular electromagnetic waves that follow
from Maxwell's equations turned into the wave equation. These waves all
satisfy each of Maxwell's equations separately.

For example, from Gauss' Laws we see e.g. that:

(9.26)

or (dividing out nonzero terms and then repeating the reasoning for
):

(9.27)

Which basically means for a real unit vector that
and
are perpendicular to , the
direction of propagation! A plane electromagnetic wave is therefore a
transverse wave. This seems like it is an important thing to
know, and is not at all a mathematical conclusion of the wave
equation per se.

Repeating this sort of thing using one of the the curl eqns (say,
Faraday's law) one gets:

(9.28)

(the cancels,
). This means
that and have the same phase if is real9.4

If is a real unit vector in 3-space, then we can
introduce three real, mutually orthogonal unit vectors
such that
and use them to
express the field strengths:

(9.29)

and

(9.30)

where and are constants that may be complex. It is worth
noting that

(9.31)

have the same dimensions and that the magnitude of the electric field is
greater than that of the magnetic field to which it is coupled via
Maxwell's Equations by a factor of the speed of light in the medium, as
this will be used a lot in electrodynamics.

We have carefully chosen the polarization directions so that the
(time-averaged) Poynting vector for any particular component pair
points in the direction of propagation,
:

(9.32)

(9.33)

(9.34)

(9.35)

Note well the combination
, as it will
occur rather frequently in our algebra below, so much so that we will
give it a name of its own later. So much for the ``simple''
monochromatic plane wave propagating coherently in a dispersionless
medium.

Now, kinky as it may seem, there is no real9.5 reason
that
cannot be complex (while remains real!)
As an exercise, figure out the complex vector of your choice such that

(9.36)

Did you get that? What, you didn't actually try? Seriously, you're
going to have to at least try the little mini-exercises I suggest
along the way to get the most out of this book.

Of course, I didn't really expect for you to work it out on such a
sparse hint, and besides, you gotta save your strength for the real
problems later because you'll need it then. So this time, I'll
work it out for you. The hint was, pretend that is
complex. Then it can be written as:

(9.37)

(9.38)

(9.39)

So, must be orthogonal to and the
difference of their squares must be one. For example:

(9.40)

works, as do infinitely more More generally (recalling the properties of
hyberbolics functions):

(9.41)

where the unit vectors are orthogonal should work for any .

Thus the most general such that
is

(9.42)

where (sigh) and are again, arbitrary complex constants. Note that if
is complex, the exponential part of the fields becomes:

(9.43)

This inhomogeneous plave wave exponentially grows or decays in
some direction while remaining a ``plane wave'' in the other
(perpendicular) direction.

Fortunately, nature provides us with few sources and associated media
that produce this kind of behavior (imaginary ? Just
imagine!) in electrodynamics. So let's forget it for the moment, but
remember that it is there for when you run into it in field theory, or
mathematics, or catastrophe theory.

We therefore return to a more mundane and natural discussion of the
possible polarizations of a plane wave when is a real unit vector, continuing the reasoning above before our little
imaginary interlude.